同构于三阶及四阶Hessain的两个变元的C~∞函数芽

C~∞ FUNCTION GERMS OF TWO VARIABLES OF ISOMORPHIC TO ITS CUBIC HESSAIN OR HESSAIN OF DEGREE 4

设En是在0∈Rn的C∞函数芽环，M是En中唯一的极大理想．如果f∈M2且其二阶Hessain是非退化的，则f同构于它的二阶Hessain是非退化的，则f同构于它的二阶Hessain，这就是著名的Morse引理．本文将讨论两个变元的C∞函数芽，得到：（1）若f∈M3Exy，且其三阶Hessain是非退化的，则f同构于它的三阶Hessain．（2）若f∈MEXy，其四阶Hessain是非退化的，则f同构于它的四阶Hessain．显然，这是Morse引理的一种推广．

Assame that En is the ring of the C ∞ functions germs at 0 ∈Rn, M is the unique maximal ideal in En. If f∈M2 and its quadratic Hessain is non-degenerate, then f is isomorphic to its quadratic Hessain. This is famous Morse lamma. In this paper, We will discuss C ∞function germs in two variables. The results show that f (1) If f∈M3 Exy and its cubic Hessain is non-degenerate, then f is isomorphic to its cubic Hessain. (2) If f∈W4 Exy and Its Hessain of degree 4 is non-degenerate, then f is isomorphic to its Hessain of degree 4. Obviously, this is a generalization of Morse lemma.